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Jun 30, 2018
06/18

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Tarik Aougab

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Let $S_{g}$ denote the genus $g$ closed orientable surface. For $k\in \mathbb{N}$, a $k$-system is a collection of pairwise non-homotopic simple closed curves such that no two intersect more than $k$ times. Juvan-Malni\v{c}-Mohar \cite{Ju-Mal-Mo} showed that there exists a $k$-system on $S_{g}$ whose size is on the order of $g^{k/4}$. For each $k\geq 2$, We construct a $k$-system on $S_{g}$ with on the order of $g^{\lfloor (k+1)/2 \rfloor +1}$ elements. The $k$-systems we construct behave well...

Topics: Mathematics, Combinatorics, Geometric Topology

Source: http://arxiv.org/abs/1403.5123

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Sep 22, 2013
09/13

by
Tarik Aougab

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Let $\Omega=(\omega_{j})_{j\in I}$ be a collection of pairwise non-isotopic simple closed curves on the closed, orientable, genus $g$ surface $S_{g}$, such that $\omega_{i}$ and $\omega_{j}$ intersect exactly once for $i\neq j$. It was recently demonstrated by Malestein, Rivin, and Theran that the cardinality of such a collection is no more than $2g+1$. In this paper, we show that for $g\geq 3$, there exists at least two such collections with this maximum size up to the action of the mapping...

Source: http://arxiv.org/abs/1210.2797v1

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Jun 28, 2018
06/18

by
Tarik Aougab

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Let $S$ be an orientable surface with negative Euler characteristic. For $k \in \mathbb{N}$, let $\mathcal{C}_{k}(S)$ denote the $\textit{k-curve graph}$, whose vertices are isotopy classes of essential simple closed curves on $S$, and whose edges correspond to pairs of curves that can be realized to intersect at most $k$ times. The theme of this paper is that the geometry of Teichm\"uller space and of the mapping class group captures local combinatorial properties of $\mathcal{C}_{k}(S)$....

Topics: Combinatorics, Mathematics, Geometric Topology

Source: http://arxiv.org/abs/1508.00502

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Sep 23, 2013
09/13

by
Tarik Aougab

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Let $\mathcal{C}(S_{g,p})$ denote the curve complex of the closed orientable surface of genus $g$ with $p$ punctures. Masur-Minksy and subsequently Bowditch showed that $\mathcal{C}(S_{g,p})$ is $\delta$-hyperbolic for some $\delta=\delta(g,p)$. In this paper, we show that there exists some $\delta>0$ independent of $g,p$ such that the curve graph $\mathcal{C}_{1}(S_{g,p})$ is $\delta$-hyperbolic. Furthermore, we use the main tool in the proof of this theorem to show uniform boundedness of...

Source: http://arxiv.org/abs/1212.3160v2

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Jun 26, 2018
06/18

by
Tarik Aougab; Jonah Gaster

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Let $S_g$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of collections of $2g+1$ simple closed curves on $S_g$ which pairwise intersect exactly once, extending a result of the first author and further answering a question of Malestein-Rivin-Theran. To distinguish such collections up to the action of the mapping class group, we analyze their dual cube complexes in the sense of Sageev. In particular, we show that for any even $k$...

Topics: Mathematics, Geometric Topology

Source: http://arxiv.org/abs/1502.00311

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Jun 29, 2018
06/18

by
Tarik Aougab; Juan Souto

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Let $S$ be a closed orientable hyperbolic surface, and let $\mathcal{O}(K,S)$ denote the number of mapping class group orbits of curves on $S$ with at most $K$ self-intersections. Building on work of Sapir [16], we give upper and lower bounds for $\mathcal{O}(K,S)$ which are both exponential in $\sqrt{K}$.

Topics: Geometric Topology, Mathematics

Source: http://arxiv.org/abs/1606.06067

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Jun 28, 2018
06/18

by
Tarik Aougab; Samuel J. Taylor

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Given $\phi$ a pseudo-Anosov map, let $\ell_\mathcal{T}(\phi)$ denote the translation length of $\phi$ in the Teichm\"uller space, and let $\ell_\mathcal{C}(\phi)$ denote the stable translation length of $\phi$ in the curve graph. Gadre--Hironaka--Kent--Leininger showed that, as a function of Euler characteristic $\chi(S)$, the minimal possible ratio $\tau(\phi) = \frac{\ell_\mathcal{T}(\phi)}{\ell_\mathcal{C}(\phi)}$ is $\log(|\chi(S)|)$, up to uniform additive and multiplicative...

Topics: Mathematics, Geometric Topology

Source: http://arxiv.org/abs/1510.00995

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Sep 18, 2013
09/13

by
Tarik Aougab; Peter A. Storm

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Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n>3. This paper verifies this conjecture for a specific example based on the 4-dimensional hyperbolic 120-cell.

Source: http://arxiv.org/abs/0809.3568v1

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Jun 30, 2018
06/18

by
Tarik Aougab; Ian Biringer; Jonah Gaster

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We prove that there is an algorithm to determine if a given finite graph is an induced subgraph of a given curve graph.

Topics: Geometric Topology, Mathematics

Source: http://arxiv.org/abs/1702.04757

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Jun 29, 2018
06/18

by
Tarik Aougab; Ian Biringer; Jonah Gaster

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Przytycki has shown that the size $\mathcal{N}_{k}(S)$ of a maximal collection of simple closed curves that pairwise intersect at most $k$ times on a topological surface $S$ grows at most as a polynomial in $|\chi(S)|$ of degree $k^{2}+k+1$. In this paper, we narrow Przytycki's bounds by showing that $$ \mathcal{N}_{k}(S) =O \left( \frac{ |\chi|^{3k}}{ ( \log |\chi| )^2 } \right) , $$ In particular, the size of a maximal 1-system grows sub-cubically in $|\chi(S)|$. The proof uses a circle...

Topics: Geometric Topology, Mathematics

Source: http://arxiv.org/abs/1610.06514

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Jun 29, 2018
06/18

by
Tarik Aougab; Jonah Gaster; Priyam Patel; Jenya Sapir

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Let $\gamma$ be an essential closed curve with at most $k$ self-intersections on a surface $\mathcal{S}$ with negative Euler characteristic. In this paper, we construct a hyperbolic metric $\rho$ for which $\gamma$ has length at most $M \cdot \sqrt{k}$, where $M$ is a constant depending only on the topology of $\mathcal{S}$. Moreover, the injectivity radius of $\rho$ is at least $1/(2\sqrt{k})$. This yields linear upper bounds in terms of self-intersection number on the minimum degree of a...

Topics: Geometric Topology, Mathematics

Source: http://arxiv.org/abs/1603.06303